Rigidity for Markovian maximal couplings of elliptic diffusions
Rigidity for Markovian maximal couplings of elliptic diffusions
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DOI：
10.1007/s00440-016-0706-4
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发表时间：
2016-04
影响因子：
2
通讯作者：
Sayantan Banerjee;W. Kendall
中科院分区：
文献类型：
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作者：
Sayantan Banerjee;W. Kendall
最大耦合是马尔可夫过程的（概率）耦合，使得耦合时间的尾概率一致达到总变差下界（Aldous界）。马尔可夫（或浸入）耦合是由策略定义的耦合，其中任何一个过程都不允许在进行下一次转换之前查看另一个过程的未来。马尔可夫耦合通常比一般耦合更容易构造和分析，并且在概率和分析的许多分支中发挥重要作用。Hsu和Sturm在2007年的预印本中证明了布朗运动的反射耦合是从两个不同点开始的布朗运动的唯一马尔可夫最大耦合（MMC）。后来，Kuwada（Electron J Probab 14（25），633-662，2009）证明了黎曼流形上布朗运动的MMC的存在性强制了流形上反射结构的存在性。在这项工作中，我们调查适当的经常椭圆形扩散流形上，并显示如何考虑的扩散几何（包括尺寸的等距群和流量的等距）是根本的分类的空间和发电机的扩散MMC存在，特别是当MMC也持有下局部扰动的起点耦合扩散。我们还描述了这样的扩散的killing向量场（等距群的生成元）和膨胀向量场（缩放对称群的生成元）。这允许一个完整的表征，这些可能的流形和它们的扩散，存在一个MMC下的耦合扩散的起始点的局部扰动。例如，在时间齐次的情况下，它表明，唯一可能出现的流形是欧几里得空间，双曲空间和超球面。此外，允许的漂移只能从这些空间的旋转等距（和膨胀，在欧几里德的情况下）得出。在这个意义上，几何刚性现象是有效的。
Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm, in a preprint circulating in 2007, but later published in 2013, proved that thereflection-couplingof Brownian motion is theunique Markovian maximal coupling(MMC) of Brownian motions starting from two different points. Later, Kuwada (Electron J Probab 14(25), 633–662, 2009) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms ofKilling vectorfields(generators of isometry groups) anddilation vectorfields(generators of scaling symmetry groups). This permits a complete characterization of those possible manifolds and their diffusions for which there exists a MMC under local perturbations of the starting points of the coupled diffusions. For example, in the time-homogeneous case it is shown that the only possible manifolds that may arise are Euclidean space, hyperbolic space and the hypersphere. Moreover the permissible drifts can then derive only from rotation isometries of these spaces (and dilations, in the Euclidean case). In this sense, a geometric rigidity phenomenon holds good.